q-Integral

The q-analog of integration is given by

(1)

which reduces to

(2)

in the case  (Andrews 1986 p. 10).

Special cases include

			(3)
			(4)
			(5)
			(6)

A specific case gives

(7)

where  is the q-gamma function and  is a doubly periodic sigma function. If , the integral reduces to

(8)

REFERENCES:

Andrews, G. E. q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., 1986.

Jackson, F. H. "q-Definite Integrals." Quart. J. Math. 41, 163, 1910.

Jackson, F. H. "The q-Integral Analogous to Borel's Integral." Mess. Math. 47, 57-64, 1917.